Step |
Hyp |
Ref |
Expression |
1 |
|
prdslmodd.y |
|
2 |
|
prdslmodd.s |
|
3 |
|
prdslmodd.i |
|
4 |
|
prdslmodd.rm |
|
5 |
|
prdslmodd.rs |
|
6 |
|
eqidd |
|
7 |
|
eqidd |
|
8 |
|
fex |
|
9 |
4 3 8
|
syl2anc |
|
10 |
1 2 9
|
prdssca |
|
11 |
|
eqidd |
|
12 |
|
eqidd |
|
13 |
|
eqidd |
|
14 |
|
eqidd |
|
15 |
|
eqidd |
|
16 |
|
lmodgrp |
|
17 |
16
|
ssriv |
|
18 |
|
fss |
|
19 |
4 17 18
|
sylancl |
|
20 |
1 3 2 19
|
prdsgrpd |
|
21 |
|
eqid |
|
22 |
|
eqid |
|
23 |
|
eqid |
|
24 |
2
|
adantr |
|
25 |
3
|
elexd |
|
26 |
25
|
adantr |
|
27 |
4
|
adantr |
|
28 |
|
simprl |
|
29 |
|
simprr |
|
30 |
5
|
adantlr |
|
31 |
1 21 22 23 24 26 27 28 29 30
|
prdsvscacl |
|
32 |
31
|
3impb |
|
33 |
4
|
ffvelrnda |
|
34 |
33
|
adantlr |
|
35 |
|
simplr1 |
|
36 |
5
|
fveq2d |
|
37 |
36
|
adantlr |
|
38 |
35 37
|
eleqtrrd |
|
39 |
2
|
ad2antrr |
|
40 |
25
|
ad2antrr |
|
41 |
4
|
ffnd |
|
42 |
41
|
ad2antrr |
|
43 |
|
simplr2 |
|
44 |
|
simpr |
|
45 |
1 21 39 40 42 43 44
|
prdsbasprj |
|
46 |
|
simplr3 |
|
47 |
1 21 39 40 42 46 44
|
prdsbasprj |
|
48 |
|
eqid |
|
49 |
|
eqid |
|
50 |
|
eqid |
|
51 |
|
eqid |
|
52 |
|
eqid |
|
53 |
48 49 50 51 52
|
lmodvsdi |
|
54 |
34 38 45 47 53
|
syl13anc |
|
55 |
|
eqid |
|
56 |
1 21 39 40 42 43 46 55 44
|
prdsplusgfval |
|
57 |
56
|
oveq2d |
|
58 |
1 21 22 23 39 40 42 35 43 44
|
prdsvscafval |
|
59 |
1 21 22 23 39 40 42 35 46 44
|
prdsvscafval |
|
60 |
58 59
|
oveq12d |
|
61 |
54 57 60
|
3eqtr4d |
|
62 |
61
|
mpteq2dva |
|
63 |
2
|
adantr |
|
64 |
25
|
adantr |
|
65 |
41
|
adantr |
|
66 |
|
simpr1 |
|
67 |
20
|
adantr |
|
68 |
|
simpr2 |
|
69 |
|
simpr3 |
|
70 |
21 55
|
grpcl |
|
71 |
67 68 69 70
|
syl3anc |
|
72 |
1 21 22 23 63 64 65 66 71
|
prdsvscaval |
|
73 |
31
|
3adantr3 |
|
74 |
2
|
adantr |
|
75 |
25
|
adantr |
|
76 |
4
|
adantr |
|
77 |
|
simprl |
|
78 |
|
simprr |
|
79 |
5
|
adantlr |
|
80 |
1 21 22 23 74 75 76 77 78 79
|
prdsvscacl |
|
81 |
80
|
3adantr2 |
|
82 |
1 21 63 64 65 73 81 55
|
prdsplusgval |
|
83 |
62 72 82
|
3eqtr4d |
|
84 |
2
|
ad2antrr |
|
85 |
25
|
ad2antrr |
|
86 |
41
|
ad2antrr |
|
87 |
|
simplr1 |
|
88 |
|
simplr3 |
|
89 |
|
simpr |
|
90 |
1 21 22 23 84 85 86 87 88 89
|
prdsvscafval |
|
91 |
|
simplr2 |
|
92 |
1 21 22 23 84 85 86 91 88 89
|
prdsvscafval |
|
93 |
90 92
|
oveq12d |
|
94 |
33
|
adantlr |
|
95 |
36
|
adantlr |
|
96 |
87 95
|
eleqtrrd |
|
97 |
91 95
|
eleqtrrd |
|
98 |
1 21 84 85 86 88 89
|
prdsbasprj |
|
99 |
|
eqid |
|
100 |
48 49 50 51 52 99
|
lmodvsdir |
|
101 |
94 96 97 98 100
|
syl13anc |
|
102 |
5
|
adantlr |
|
103 |
102
|
fveq2d |
|
104 |
103
|
oveqd |
|
105 |
104
|
oveq1d |
|
106 |
93 101 105
|
3eqtr2rd |
|
107 |
106
|
mpteq2dva |
|
108 |
2
|
adantr |
|
109 |
25
|
adantr |
|
110 |
41
|
adantr |
|
111 |
|
simpr1 |
|
112 |
|
simpr2 |
|
113 |
|
eqid |
|
114 |
23 113
|
ringacl |
|
115 |
108 111 112 114
|
syl3anc |
|
116 |
|
simpr3 |
|
117 |
1 21 22 23 108 109 110 115 116
|
prdsvscaval |
|
118 |
80
|
3adantr2 |
|
119 |
4
|
adantr |
|
120 |
1 21 22 23 108 109 119 112 116 102
|
prdsvscacl |
|
121 |
1 21 108 109 110 118 120 55
|
prdsplusgval |
|
122 |
107 117 121
|
3eqtr4d |
|
123 |
92
|
oveq2d |
|
124 |
|
eqid |
|
125 |
48 50 51 52 124
|
lmodvsass |
|
126 |
94 96 97 98 125
|
syl13anc |
|
127 |
102
|
fveq2d |
|
128 |
127
|
oveqd |
|
129 |
128
|
oveq1d |
|
130 |
123 126 129
|
3eqtr2rd |
|
131 |
130
|
mpteq2dva |
|
132 |
|
eqid |
|
133 |
23 132
|
ringcl |
|
134 |
108 111 112 133
|
syl3anc |
|
135 |
1 21 22 23 108 109 110 134 116
|
prdsvscaval |
|
136 |
1 21 22 23 108 109 110 111 120
|
prdsvscaval |
|
137 |
131 135 136
|
3eqtr4d |
|
138 |
5
|
fveq2d |
|
139 |
138
|
adantlr |
|
140 |
139
|
oveq1d |
|
141 |
33
|
adantlr |
|
142 |
2
|
ad2antrr |
|
143 |
25
|
ad2antrr |
|
144 |
41
|
ad2antrr |
|
145 |
|
simplr |
|
146 |
|
simpr |
|
147 |
1 21 142 143 144 145 146
|
prdsbasprj |
|
148 |
|
eqid |
|
149 |
48 50 51 148
|
lmodvs1 |
|
150 |
141 147 149
|
syl2anc |
|
151 |
140 150
|
eqtr3d |
|
152 |
151
|
mpteq2dva |
|
153 |
2
|
adantr |
|
154 |
25
|
adantr |
|
155 |
41
|
adantr |
|
156 |
|
eqid |
|
157 |
23 156
|
ringidcl |
|
158 |
2 157
|
syl |
|
159 |
158
|
adantr |
|
160 |
|
simpr |
|
161 |
1 21 22 23 153 154 155 159 160
|
prdsvscaval |
|
162 |
1 21 153 154 155 160
|
prdsbasfn |
|
163 |
|
dffn5 |
|
164 |
162 163
|
sylib |
|
165 |
152 161 164
|
3eqtr4d |
|
166 |
6 7 10 11 12 13 14 15 2 20 32 83 122 137 165
|
islmodd |
|